On the Number of Iterations for Dantzig-Wolfe Optimization and - - PowerPoint PPT Presentation

On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms Phil Klein

Brown University

Neal Young

Dartmouth College

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simple multicommodity flow problem

P = {si → ti paths}. Route at least 1 total unit of flow,

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respecting capacity constraint c (0 < c < 1).

simple multicommodity flow algorithm

P = {si → ti paths}. Repeat for T iterations:

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Route 1/T units of flow on min-cost path in P, where cost of edge e is (1 + ǫ)T flow(e).

performance guarantee

THM: If flow of congestion c exists, then algorithm returns flow of congestion c(1 + ǫ) provided T ≥ 3 ln(m) c ǫ2 .

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generic packing problem

input: real matrix A, vector b, generic polytope P

• utput: x ∈ P such that Ax ≤ b.

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generic packing problem

input: real matrix A, vector b, generic polytope P

• racle for P: given vector c returns arg min

x∈P c · x.

ǫ-approximate solution: x ∈ P such that Ax ≤ (1 + ǫ)b.

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Lagrangian Relaxation

idea: replace constraints by costs 1950: Ford, Fulkerson Reduced multicommod. flow to iterated min-cost flow. 1960: Dantzig, Wolfe Generalized to generic packing problem. 1970: Held, Karp Reduced TSP l.b. to iterated min. 1-spanning-tree. 1990: Shahrokhi, Matula Multicommodity flow, guaranteed convergence rate. . . . 1995: Plotkin, Shmoys, Tardos Generalized to generic packing problem.

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# Iterations prop. to ρ ln(m)/ǫ2 ρ = “width”, m =# constraints. . . .

main result: a lower bound

THM: Any ǫ-approximation algorithm for the generic packing problem requires a number of iterations prop. to T = ρ ln(m)/ǫ2 for sufficiently large m.

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proof idea (ρ = 2):

Reduce to question about two-player zero-sum matrix games. value(M) = min

x∈P max j

(M x)j, where P = {x ≥ 0 :

i xi = 1}.

THM: Let M be a random matrix in {0, 1}m×√m. With high probability, every m × T submatrix B of M has value(B) > (1 + ǫ)value(M) where T = Ω(ln(m)/ǫ2). COROLLARY: At least T oracle calls to know value(M) within 1 + ǫ.

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underlying idea:

Show m × T submatrix has high value with high probability:

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Discrepancy theory.